I would link to the relevant question also to show this question is related to a textual question (I know the context, but others may not and assume this is just a Hebrew language question, which would be off topic).
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user1985Feb 17 '13 at 22:44

While Shabbat does fall on the seventh day of the week, Strong's doesn't note a linguistic connection between the two words nor have I ever learned one (for what that's worth). As far as I can tell they are just two similar-looking words.

It appears from a cursory inspection of tractate Shabbat in the Babylonian talmud that Aramaic uses the same roots for these words. (This is as expected, but I did my best to verify. My Aramaic is weaker than my Hebrew.)

I do not know about Greek.

Please note: This answer was written for a neutral, academic audience
and is not intended to be interpreted in the context of a religious
belief or doctrine.

While writing this I tried to research how often words that share two of three root letters are related (if ever), but I didn't find anything concrete before I gave it up as a tangent to this question. Maybe that would make a good question on its own.
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Gone QuietFeb 17 '13 at 21:46

Someone else said, "The words have two distinct roots. The root for Sabbath (שבת) means "rest". It is not directly etymologically related to the root for seven (שבע) in either language. However, the notion of some distant etymological relationship has occurred to me; see for example the entry for "seven" in this Akkadian dictionary. (I don't know much about Akkadian, so it could be that the "t" or "s" in some variants of the word are not derived from the root)."
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user2027Feb 18 '13 at 2:00

Hebrew roots shows seemingly unconnected words that one will learn are divinely connected. Shabbat shares roots with "sheva" which means "7" and sheb'ou'aw which means "oath" and "shebet" which is a branch or scion or stick for writing or ruling, and shaba, which mean to have plenty. By the same token, with Hebrew being a picture language, "Torah," an archery term which means "to hit the mark," and has the same root as the word for teacher, moreh (מורה). A moreh is one who imparts instruction to his/her students. The second important word is parent, horeh (הורה). This indicates to us that one of the primary roles for a parent is to teach and instruct the child.

Hello Rivkah, and welcome to Biblical Hermeneutics! We appreciate your thoughts here, but it isn’t self evident that words sharing two root letters are connected. Both of the other answers here have indicated that they are not. If you’d like to argue that they are connected, it would be helpful to cite a reliable source that explains the linguistic connection. Thanks.
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Susan♦Feb 21 at 22:29